Problem: Simplify and expand the following expression: $ \dfrac{2}{2a - 16}- \dfrac{5}{5a + 30}+ \dfrac{3}{a^2 - 2a - 48} $
Explanation: First find a common denominator by finding the least common multiple of the denominators. Try factoring the denominators. We can factor a $2$ out of denominator in the first term: $ \dfrac{2}{2a - 16} = \dfrac{2}{2(a - 8)}$ We can factor a $5$ out of denominator in the second term: $ \dfrac{5}{5a + 30} = \dfrac{5}{5(a + 6)}$ We can factor the quadratic in the third term: $ \dfrac{3}{a^2 - 2a - 48} = \dfrac{3}{(a - 8)(a + 6)}$ Now we have: $ \dfrac{2}{2(a - 8)}- \dfrac{5}{5(a + 6)}+ \dfrac{3}{(a - 8)(a + 6)} $ The least common multiple of the denominators is: $ 10(a - 8)(a + 6)$ In order to get the first term over $10(a - 8)(a + 6)$ , multiply by $\dfrac{5(a + 6)}{5(a + 6)}$ $ \dfrac{2}{2(a - 8)} \times \dfrac{5(a + 6)}{5(a + 6)} = \dfrac{10(a + 6)}{10(a - 8)(a + 6)} $ In order to get the second term over $10(a - 8)(a + 6)$ , multiply by $\dfrac{2(a - 8)}{2(a - 8)}$ $ \dfrac{5}{5(a + 6)} \times \dfrac{2(a - 8)}{2(a - 8)} = \dfrac{10(a - 8)}{10(a - 8)(a + 6)} $ In order to get the third term over $10(a - 8)(a + 6)$ , multiply by $\dfrac{10}{10}$ $ \dfrac{3}{(a - 8)(a + 6)} \times \dfrac{10}{10} = \dfrac{30}{10(a - 8)(a + 6)} $ Now we have: $ \dfrac{10(a + 6)}{10(a - 8)(a + 6)} - \dfrac{10(a - 8)}{10(a - 8)(a + 6)} + \dfrac{30}{10(a - 8)(a + 6)} $ $ = \dfrac{ 10(a + 6) - 10(a - 8) + 30} {10(a - 8)(a + 6)} $ Expand: $ = \dfrac{10a + 60 - 10a + 80 + 30}{10a^2 - 20a - 480} $ $ = \dfrac{170}{10a^2 - 20a - 480}$ Simplify: $ = \dfrac{17}{a^2 - 2a - 48}$